8. Practice questions and answers. Add your own questions

1. What were the first numbers?	The numbers, most probably invented first were one and two. Some primitive races found it much of a strain to enter ! into further refinements and have in their language only three words of number — one, two and many. The invention of the process of counting objects in succession (that is "counting by ones") led our ancestors to develop the numbers 1, 2, 3, 4 and so on. These are called the natural numbers. From the experience with counting objects they developed next the idea of the sum of two numbers.
2. What is a fraction?	Early in history, practical problems of measurement involving the division of things into equal parts led to the invention of fractions. The ancient Greeks were acquainted with fractions but regarded a fraction as a relation between two numbers. From the modern point of view, a fraction is considered as a single number because it does have many important properties in common with natural numbers, such as the associative, commutative and distributive laws. The natural numbers and the fractions satisfied the practical needs of man for many thousands of years and were the only ones dealt with until comparatively modern times. The system of fractions is closed under the operation of addition, multiplication and division, but not subtraction. It is impossible to subtract a larger fraction from a smaller one, or even a fraction from itself. This situation is remedied by the extending the idea of number in a fundamental way.
3. What do we mean by directed numbers?	Corresponding to each number already in existence, such as 3, two new symbols or marks are added +3 and -3. Thus, we speak of positive and negative whole numbers or integers and fractions. Zero is neither positive nor negative. All these new symbols (+3, +'/2) are called directed or signed numbers because they may be directed geometrically (as points on a line) in terms of direction. The plus and minus signs in' the symbols (+3, -3) are not intended to indicate the operations of addition or substraction, but are merely marks to distinguish one direction from another. The idea of a negative number struggled for recognition for centuries and was received with great reluctance as late as the early years of the seventeenth century even by mathematicians. They were often called false or fictitious numbers.
4. What is a rational number?	All the numbers introduced so far, namely, zero, the positive and negative integers and fractions (and no others!) are called rational numbers. The word "rational" does not mean reasonable. It comes from the word "ratio": every rational number can be expressed as a quotient or ratio of two integers. Division by zero must be excluded in this system.
5. What is an irrational number?	Our system of rational numbers contains no number which can represent the length of the hypotenuse of an isosceles right triangle whose leg is of unit length. The word "irrational" means not expressible as a ratio of two integers. The strange story of is dated back to Pythagoras who had been so much disturbed by his revelation that tried to suppress the information lest it discredit mathematicians in the eyes of the general public. The difficulties involved were not straightened out with complete logical rigor until 1870 when C. Cantor and R.Dedekind did it independently of each other. Recall the usual decimal notation for numbers. If a deci­mal terminates, it represents a rational number, if not irrational numbers, is approximated by rational numbers and can be so approximated as closely as we wish by carrying the process far enough. This is one essential characteristic of irrational numbers. The decimal expression forcan neither terminate nor be periodic.
6. What is a real number?	We shall assume that the irrational numbers fill up all the "gaps" left in our straight line, so that every point on the line has a number attached to it. These numbers are called real. Every real number can be expressed as a decimal either terminating or not and conversely. The system of real numbers is subdivided into rational and irrational numbers.
7. What is a complex number ?	Consider any negative number, say, -4.A square root of -A is x, x^2 = -4. Such a number x cannot be found in our extended system of real numbers. Any symbol of the form where b stands for a real number(0) andn for any negative number will be called a pure imaginary number (4). If we add a real number a to it, we get a peculiar hybrid which is called a complex number (3 + 5). The complex numbers include all the previous kinds of numbers as special cases. Ifb 0, a complex number is called imaginary. We may summarize the entire number system as it now stands: Complex numbers Imaginary b0 Pure imaginary (a = 0, 0 b ^ 0) Real (b = 0) Irrational Negative integers Rational Zero Integers Positive integers or natural numbers Don't think that the word "imaginary" means that these numbers are mystical or unreal in the everyday sense of the word, or that "complex" means complicated. Imaginary numbers have very "real" applications in many branches of science. The system composed of real numbers and pure imaginary numbers is not closed under addition since the sum of a real number and a pure imaginary number is neither real nor pure imaginary. Hence, the system of complex numbers was invented.
8. What are new number systems?	In the 19th century several new number-systems were invented. Of these modern systems three are particularly noteworthy: quaternions (triplets), matrices, and transfinite numbers.
9. What new concepts has axiomatic inquiry brought forth?	Axiomatic inquiry has brought forth new concepts such as rings, fields, groups, and
10. What is one of greatest accomplishments in number theory?	It is the statement that the real number system is a "complete ordered field".